Existence and stability of standing pulses in neural networks

This dissertation studies a one dimensional neural network rate model that supports localized self-sustained solutions. These solutions could be an analog for working memory in the brain. Working memory refers to the temporary storage of information necessary for performing different mental tasks. Cortical neurons that show persistent activity are observed in animals during working memory tasks. The physical process underlying this persistent activity could be due to self-sustained network activity of the neurons in the brain.The term `bump' has been coined to imply a spatially localized persistent activity state that is sustained internally by a network of neurons. Many researchers have analyzed the bump state using Firing rate models with either the Heaviside gain function or a saturating sigmoidal one. These gain functions imply that neurons begin to fire once their synaptic input reaches threshold, and the firing rate saturates to a maximal value almost immediately. However, cortical neurons that exhibit persistent activity usually are well below their maximal attainable rate. To resolve this paradox, I study a single population rate model using a biophysically relevant firing rate function.I consider the existence and the stability of standing single-pulse solutions of an integro-diferential neural network equation. In this network, the synaptic coupling has local excitatory coupling with distal lateral inhibition and the non-saturating gain function is piece-wise linear. A standing pulse solution of this network is a synaptic input pattern that supports a bump state. I show that the existence condition for single-pulses of the integro-differential equation can be reducedto the solution of an algebraic system. With this condition, I map out the shape of the pulsesfor different coupling weights and gains. By a similar approach, I also find the conditions for the existences of dimple-pulses and double-pulses. For a fixed gain and connectivity, there are at least two single-pulse solutions - a "large" one and a "small" one. However, more than two single-pulses can coexist depending on the parameter range. To have standing single-pulses, the gain function and synaptic coupling are both important.I also derive a stability criteria for the standing pulse solutions. I show that the large pulse is stable and the small pulse is unstable. If there are more than two pulse solutions coexisting, the first pulse is the small one and it is unstable. The second one is a large stable pulse. The third pulse is wider than the second one and it is unstable. More importantly, the second single-pulse (which could be a dimple pulse) is bistable with the "all-off" state. The stable pulse represents the memory. When the network is switched to the "all-off" state, the memory is erased.